Velocity control in Parkinson's disease: a quantitative analysis of isochrony in scribbling movements

An experiment was conducted to contrast the motor performance of three groups (N=20) of participants: (1) patients with confirmed Parkinson Disease (PD) diagnose; (2) age-matched controls; (3) young adults. The task consisted of scribbling freely for 10s within circular frames of different sizes. Comparison among groups focused on the relation between the figural elements of the trace (overall size and trace length) and the velocity of the drawing movements. Results were analysed within the framework of previous work on normal individuals showing that instantaneous velocity of drawing movements depends jointly on trace curvature (Two-thirds Power Law) and trace extent (Isochrony principle). The motor behaviour of PD patients exhibited all classical symptoms of the disease (reduced average velocity, reduced fluency, micrographia). At a coarse level of analysis both isochrony and the dependence of velocity on curvature, which are supposed to reflect cortical mechanisms, were spared in PD patients. Instead, significant differences with respects to the control groups emerged from an in-depth analysis of the velocity control suggesting that patients did not scale average velocity as effectively as controls. We factored out velocity control by distinguishing the influence of the broad context in which movement is planned—i.e. the size of the limiting frames—from the influence of the local context—i.e. the linear extent of the unit of motor action being executed. The balance between the two factors was found to be distinctively different in PD patients and controls. This difference is discussed in the light of current theorizing on the role of cortical and sub-cortical mechanisms in the aetiology of PD. We argue that the results are congruent with the notion that cortical mechanisms are responsible for generating a parametric template of the desired movement and the BG specify the actual spatio-temporal parameters through a multiplicative gain factor acting on both size and velocit

When One Size Does Not Fit All: A Simple Statistical Method to Deal with Across-Individual Variations of Effects

In science, it is a common experience to discover that although the investigated effect is very clear in some individuals, statistical tests are not significant because the effect is null or even opposite in other individuals. Indeed, t-tests, Anovas and linear regressions compare the average effect with respect to its inter-individual variability, so that they can fail to evidence a factor that has a high effect in many individuals (with respect to the intra-individual variability). In such paradoxical situations, statistical tools are at odds with the researcher’s aim to uncover any factor that affects individual behavior, and not only those with stereotypical effects. In order to go beyond the reductive and sometimes illusory description of the average behavior, we propose a simple statistical method: applying a Kolmogorov-Smirnov test to assess whether the distribution of p-values provided by individual tests is significantly biased towards zero. Using Monte-Carlo studies, we assess the power of this two-step procedure with respect to RM Anova and multilevel mixed-effect analyses, and probe its robustness when individual data violate the assumption of normality and homoscedasticity. We find that the method is powerful and robust even with small sample sizes for which multilevel methods reach their limits. In contrast to existing methods for combining p-values, the Kolmogorov-Smirnov test has unique resistance to outlier individuals: it cannot yield significance based on a high effect in one or two exceptional individuals, which allows drawing valid population inferences. The simplicity and ease of use of our method facilitates the identification of factors that would otherwise be overlooked because they affect individual behavior in significant but variable ways, and its power and reliability with small sample sizes (<30–50 individuals) suggest it as a tool of choice in exploratory studies

Comparison of type II error rates in UKS test and RM Anovas

<p>. Results of a simulation study based on over one billion datasets. Each dataset represents the data of 10 individuals performing 10 trials in each of the 2 levels of a factor. Each data point was obtained by adding to the fixed central value of the level (−1/√2 or +1/√2) two random Gaussian values representing individual idiosyncrasies and trial-to-trial errors (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0039059#s4" target="_blank">Methods</a>). <i>Panel A:</i> Median probability (Z-axis) yielded by RM Anovas as a function of the standard deviations of subject-factor interaction (X-axis, rightwards) and average of 10 trial-to-trial errors (Y-axis, leftwards). <i>Panel B:</i> Median probability yielded by the UKS test for the same random data. <i>Panel C:</i> superimposition of the surfaces displayed in panel A and B. Note that in conditions when UKS test is less powerful than ANOVA (larger median p), the difference in power is never dramatic; the converse is not true. <i>Panel D:</i> 2D-isolines of the surfaces in panel C for median probabilities. 001 (red), .01 (orange), .05 (green), .10 (light blue) and .20 (dark blue). Black line: projection of the intersection of the two surfaces; RM Anova is more powerful (smaller median probability) than the UKS test for points leftwards of the black line. Note that scaling the X-axis to the SD of within-level averages of trial-to-trial errors gives a symmetrical aspect to RM Anova surface and projection.</p

Type II errors and reproducibility with heterogeneous experimental effects

<p>. Each panel displays the proportion of significant hypothetical experiments as a function of the difference <i>d</i> between the constant values of experimental effect in 2 (panels A–E) or 3 sub-populations (panel F). The lines show the proportion of significant tests in 10000 hypothetical experiments for 41 values of <i>d</i> from 0 to 8 by .2 steps for RM Anovas (continuous line) and the UKS test at both the .05 (dashed line) and.01 threshold (dotted line). The gray part of lines indicates the 0.211–0.789 range of proportion of significant tests for which the probability that two subsequent experiments yield conflicting outcomes exceeds 1/3. Each experiment consists in 10 individuals performing 8 trials in a baseline condition and in an experimental condition. Trial errors are drawn from a Gaussian distribution with parameters 0 and √8, so that the average of the experimental condition has a Gaussian distribution centered on –<i>d</i>, 0 or +<i>d</i> (Insets) with unitary variance. The proportion and center of the subpopulations varied across studies. In the first study (<i>panel A</i>), the experimental effect was set to 0 for 10% of the population, and to <i>d</i> for the remaining 90%. In the other studies (<i>Panels B–F</i>), the effects and proportions were as follows: [0, 20%; <i>d</i>, 80%]; –<i>d</i>, 10%; <i>d</i>, 90%]; [0, 40%; <i>d</i>, 60%]; –<i>d</i>, 20%; <i>d</i>, 80%]; [–<i>d</i>, 10%; 0; 20%; <i>d</i>, 70%]. For each hypothetical experiment, the 10 individual effects were drawn with replacement from a set of –<i>d</i>, 0 and +<i>d</i> values in the above proportions (for <i>d</i> = 0, the proportion of significant tests is equal to the nominal type I error rate). We conclude that when factor effects vary across individuals as modeled by a mixture of Gaussians, UKS tests yield more reproducible outcomes than RM Anovas and have lower type II errors.</p

Violations of homoscedasticity and normality assumptions in one-way Anova design: compared robustness of RM Anova and UKS test.

<p><i>Panel A:</i> Violation of equal variance assumption. Curves display trial-to-trial errors distributions in the factor levels with the smallest and largest variance for the 4 degrees of heteroscedasticity investigated in simulation studies (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0039059#s4" target="_blank">Methods</a>). The numbers under the curves indicate the average percentage of type I errors (false positives) for RM Anovas, individual Anovas and the UKS test procedure, respectively. Numbers above 5% indicate an excess of significant datasets with respect to the tests threshold (0.05). We observe that the UKS test, as the RM Anova, is robust to heterogeneity of variance. <i>Panel B:</i> Violation of normality assumption. Curves display the empirical distributions of trial-to-trial errors drawn from the following 4 distributions: gamma with k = 4; lognormal with μ = 0 and σ = 1/√2; Weibull with k = 1.2 and λ = 0.5; exponential with λ = 0.4 (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0039059#s4" target="_blank">Methods</a>). Boxes: Normal probability plots of typical residuals from an Anova applied to skewed data randomly drawn from the above distribution. For the displayed residuals (10 individuals × 3 levels × 10 repetitions with a median coefficient of correlation r), skewness is significant at the .01 threshold when r <0.9942. The numbers under the boxes indicate the across-designs average percentage of type I errors (false positives) for individual Anovas and UKS test applied to raw data or after a logarithmic transformation. Numbers above 5% indicate an excess of significant datasets with respect to the threshold used (0.05). When data is skewed, the UKS test should be used in conjunction with individual nonparametric tests (see text, Part 7), or data should be (log-)transformed.</p